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Local non-stationarity test in mean for Markov switching GARCH models: an approximate Bayesian approach

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Abstract

In order to exploit mean-reverting behavior among the price differential between two markets, one can use unit root tests to determine which pairs of assets appear to exhibit mean-reverting behavior. Since nonlinear mean reversion shares the same meaning as local stationarity, this paper proposes a Bayesian hypothesis testing to detect the presence of a local unit root in the mean equation using Markov switching GARCH models. This model incorporates a fat-tailed error distribution to analyze asymmetric effects on both the conditional mean and conditional volatility of financial time series. To implement the test, we propose a numerical approximation of the marginal likelihoods to posterior odds by using an adaptive Markov Chain Monte Carlo scheme. Our simulation study demonstrates that the approximate Bayesian test performs properly. The proposed method utilizes the daily basis between the FTSE 100 Index and Index Futures as an illustration.

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Notes

  1. See Sarno and Valente (2000) and Monoyios and Sarno (2002).

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Acknowledgments

We thank the AE and referees very much for their time and careful, extensive comments on our paper, which have led to an improved version of it. Cathy W.S. Chen’s research is funded by the Ministry of Science and Technology, Taiwan (NSC 101-2118-M-035-006-MY2, MOST 103-2118-M-035-002-MY2). Sangyeol Lee’s research is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (No. 2015R1A2A2A010003894).

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Authors and Affiliations

  1. Department of Statistics, Feng Chia University, Taichung, Taiwan

    Cathy W. S. Chen & Shu-Yu Chen

  2. Department of Statistics, Seoul National University, Seoul, Korea

    Sangyeol Lee

Authors
  1. Cathy W. S. Chen
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  2. Sangyeol Lee
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  3. Shu-Yu Chen
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Corresponding author

Correspondence to Cathy W. S. Chen.

Appendix: MCMC sampling scheme

Appendix: MCMC sampling scheme

The MCMC sampler consists of simulating each parameter group from its full conditional posterior discussed in Sect. 3. The symbol k on a parameter denotes a draw of the parameter at the kth iteration of the algorithm. We randomly select the initial value of the state vector \({\varvec{S}}\) from a first-order Markov process assuming values in \(\{1, 2\}\). We set the initial values for all parameters, i.e. \(k =0\). One iteration of the algorithm involves the following steps:

Step 1: :

Sample \({\varvec{\phi }}_i|{\varvec{y}},{\varvec{\delta }}, {\varvec{S}}, {\varvec{\theta }}_{-{\varvec{\phi }}_i}\) from its posterior in (12). Check the constraint that \({\phi }_1^{(1)} \ge {\phi }_1^{(2)}\). If this constraint is not satisfied, then we relabel the draws \(({\varvec{\phi }}_1,{\varvec{\phi }}_2)\).

Step 2: :

Sample \(\delta ^{(i)}\) from a Bernoulli distribution with success probability in (14). Count the number \(I(\delta ^{(i)}=1)\), \(i=1, 2\).

Step 3: :

Sample \(s_t| {\varvec{y}},{\varvec{\delta }},{\varvec{S}}_{-t},{\varvec{\theta }}\), \(t=2, \ldots , T\) from its posterior in (14).

Step 4: :

Sample \(p_{ii}| {\varvec{y}}, {\varvec{\delta }},{\varvec{S}}, {\varvec{\theta }}_{-p_{ii}}\) from its posterior in (18).

Step 5: :

Sample \({\varvec{\alpha }}_i|{\varvec{y}}, {\varvec{\delta }},{\varvec{S}}, {\varvec{\theta }}_{-{\varvec{\alpha }}_i};\) from its posterior in (19).

Step 6: :

Sample \(\nu | {\varvec{y}}, {\varvec{\delta }},{\varvec{S}}, {\varvec{\theta }}_{-\nu }\) from its posterior.

Step 7: :

\(k=k+1\) and go to Step 1.

These steps are repeated until \(k=N\) so that the convergence of the Markov chain is achieved, and we calculate the posterior median \(\tilde{{\varvec{\theta }}}\). The first term of Eq. (8) is the log-likelihood evaluated at \(\tilde{{\varvec{\theta }}}\). We determine the mean of \(\pi _i\) (denoted by \(\tilde{\pi }_i\)) from Step 2, which is used to compute the prior odds ratio \(\tilde{\pi }_i/(1-\tilde{\pi }_i)\). The value of \(\text{ POR }= \text{ BF }\times \tilde{\pi }_i/(1-\tilde{\pi }_i)\) to test the unit root hypotheses is then obtained.

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Chen, C.W.S., Lee, S. & Chen, SY. Local non-stationarity test in mean for Markov switching GARCH models: an approximate Bayesian approach. Comput Stat 31, 1–24 (2016). https://doi.org/10.1007/s00180-015-0624-4

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